Inferring Parameters and Structure of Latent Variable Models by Variational Bayes (1301.6676v1)

Abstract: Current methods for learning graphical models with latent variables and a fixed structure estimate optimal values for the model parameters. Whereas this approach usually produces overfitting and suboptimal generalization performance, carrying out the Bayesian program of computing the full posterior distributions over the parameters remains a difficult problem. Moreover, learning the structure of models with latent variables, for which the Bayesian approach is crucial, is yet a harder problem. In this paper I present the Variational Bayes framework, which provides a solution to these problems. This approach approximates full posterior distributions over model parameters and structures, as well as latent variables, in an analytical manner without resorting to sampling methods. Unlike in the Laplace approximation, these posteriors are generally non-Gaussian and no Hessian needs to be computed. The resulting algorithm generalizes the standard Expectation Maximization algorithm, and its convergence is guaranteed. I demonstrate that this algorithm can be applied to a large class of models in several domains, including unsupervised clustering and blind source separation.

• The paper presents a novel Variational Bayes framework that analytically approximates posterior distributions while optimally inferring latent model structures.
• It employs an EM-like algorithm to estimate both parameter and hidden variable posteriors, effectively reducing overfitting compared to maximum likelihood methods.
• Applications to mixture models and blind source separation demonstrate its capacity to automatically determine model complexity and improve separation quality.

Variational Bayes in Latent Variable Models

The paper introduces a framework for learning graphical models with latent variables, addressing key problems associated with existing methods, such as overfitting and computational tractability. The proposed approach, Variational Bayes (VB), provides an analytical approximation of the posterior distributions over model parameters, structures, and latent variables, circumventing the limitations of Maximum Likelihood (ML) and standard Bayesian methods.

Core Concepts and Methodology

Current ML methods optimize model parameters for a fixed graph structure but often lead to overfitting and suboptimal generalization. Additionally, ML struggles with learning model structures, as it favors more complex graphs that assign higher likelihoods to data, and it remains computationally feasible only for a narrow class. The Bayesian approach theoretically remedies these issues by offering a probabilistic treatment of ensembles characterized by distributions over potential parameter values and structures. However, exact computations in this framework are intractable, necessitating approximations, such as Markov chain Monte Carlo and Laplace approximation, both of which have significant limitations.

VB addresses these challenges by providing an analytical solution for computing approximate posterior distributions without sampling or assuming Gaussian posteriors. The framework approximates the true posterior by a variational posterior designed to maximize the log-likelihood lower bound, effectively penalizing complex models and allowing for optimal structure learning.

Algorithmic Implementation

The VB framework unfolds as an EM-like algorithm iterating over two main steps, handling parameter and hidden variable posteriors separately:

1. Parameter Posteriors: For a given structure, it factors across nodes, utilizing appropriate priors such as Dirichlet for discrete nodes and Normal-Wishart for continuous ones. The result is a more efficient parameter approximation than the Laplace method, producing non-trivial posteriors for any sample size.
2. Hidden Variable Posteriors: These are optimized using parametric forms that facilitate computational feasibility, even for complex models. The algorithm iteratively adjusts these using data-driven variational parameters.

VB facilitates efficient model learning by transforming complex integrations into tractable operations, thus addressing the computational bottlenecks of exact Bayesian inference.

Applications and Results

VB is applied to a variety of models, including mixture models and blind source separation:

• Mixture Models: The framework enables a flexible approach to density estimation, automatically determining the optimal number of mixture components. The VB algorithm outperforms traditional EM by circumventing the singularity issues of component covariance matrices. Results on test datasets demonstrate its capacity to accurately identify model structure, reflecting peaked posteriors over the true number of components.
• Blind Source Separation: Applied to the BSS problem, VB learns the mixing matrix and source distributions without prior knowledge of the number of sources. The algorithm shows robustness across different noise levels, correctly inferring source numbers and delivering improved separation quality.

Implications and Future Directions

VB provides a versatile and computationally efficient approach for learning complex models in AI, particularly for problems involving hidden variables and unknown structures. It has significant implications for fields requiring nuanced model understanding, such as speech recognition and sensor data analysis.

The paper hints at the potential for VB to be expanded to hierarchical models and dynamic Bayesian networks, which could further enhance its utility in real-world applications. Future research may focus on comparing VB's precision against other approximations and extending its applicability to more intricate network structures, including non-linear and temporal models.